Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation

We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers’ equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $${\mathrm{\partial }}_{t}u-D_x^{\alpha }{\mathrm{\partial }}_{x}u={\mathrm{\partial }}_x(u^2),$$ and prove that it is locally well-posed in $H^s(\mathbb{K})$, $\mathbb{K}=\mathbb{R}$ or $\mathbb{T}$, for $s>s_{\alpha }$, where \begin{equation*} s_α=\begin{cases} 1-\frac{3α}4 & $\text{for}$ \quad \frac23 \le α\le 1; \frac 32(1-α) & $\text{for}$ \quad \frac13 \le α\le \frac23; \frac 32-\fracα{1-α} & $\text{for}$ \quad 0 < α\le \frac13 . \end{cases} \end{equation*} The uniqueness is unconditional in $H^s(\mathbb{K})$ for $s>\max \{\frac{1}{2},s_{\alpha }\}$. Moreover, we obtain \emph{a priori} estimates for the solutions at the lower regularity threshold $s>{\tilde{s}}_{\alpha }$ where \begin{equation*} \widetilde{s}_α=\begin{cases} \frac 12-\frac α4 & $\text{for}$ \quad \frac23 \le α\le 1; 1-α& $\text{for}$ \quad \frac12 \le α\le \frac23; \frac 32-\fracα{1-α} & $\text{for}$ \quad 0 < α\le \frac12 . \end{cases} \end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb{K})$ for $s>s_{\alpha }$ when $\alpha >\frac{2}{3}$, and in the energy space $H^{\frac{\alpha }{2}}(\mathbb{K})$ when $\alpha >\frac{4}{5}$.